Which is correct: The rest of the staff is or are? The rest of my family is or are?

I would use the singular (is) in both examples as we are referring one "staff" and one "family", even though they may consist of many people.

The same can be said for the nouns "team", "company", "organisation" etc.

NB: Edited after Peter's comment

This is my opinion. Family and staff are collective nouns. A collective noun, as we know, may take a singular or plural verb depending on whether we see it as a unit or a collection of individuals. Therefore, if the rest of the family is moving as one, then we can say, "The rest of my family is arriving late" (this means the other members of the family are arriving together).

Fraser Orr is correct in saying that the subject is the word rest but the singularization or pluralization of the verb depends much on the specific noun that follows the abovementioned subeject. Take the following examples.

• The rest of the apples are rotten.
• The rest of the book was burned.

The use of this becomes confusing if the noun that follows is a collective noun.

Either singular or plural can be correct, especially in British English. The reason can be seen in two steps, involving two things which complicate subject-verb agreement: number-transparent nouns and collective nouns (CGEL, pp. 501-504).

^{Let me remind you that the subject is always the whole noun phrase (NP), so in your first example, the subject is the NP the rest of the staff, and in your second example, the subject is the NP the rest of my family. The question of whether you need the singular is or the plural are rest on whether the NPs the rest of the staff and the rest of my family are singular or plural.}

First step: note that rest is a number-transparent noun (CGEL, p. 350). Consider the following examples with another number-transparent noun, lot:

[1] a.  [A lot of work] was done.
      b.  [A lot of errors] were made.

In both [1a] and [1b], the subject (in brackets) is a noun phrase (NP) whose head is lot. But the number of the NP (i.e. whether the NP is singular or plural) is determined not by the head, as is normally the case, but by the number of the oblique (i.e. by the number of the complement of the preposition of): a lot of work is singular because work is singular, and so we have the singular verb was. And a lot of errors is plural because errors is plural, and so we have the plural verb were. Besides lot and rest, there are only several other clear examples of number-transparent nouns: lots, plenty, bags, heaps, loads, oodles, remainder, number, and couple (CGEL, p. 350). In the words of CGEL, a number-transparent noun allows the number of the oblique to percolate up to determine the number of the whole NP (CGEL, p. 349). In your examples, the staff and my family are the obliques, and they determine the number of the NPs the rest of the staff and the rest of my family.

So, because rest is number-transparent, the question of whether the rest of the staff is singular or plural reduces to the question of whether the staff is singular or plural. Similarly for the rest of my family and my family. And now comes step two.

Second step: even though staff and family are singular, they are collective nouns, and as such they allow optional plural override. Here is an example (CGEL, p. 501):

[2] a.  [The committee] has not yet come to a decision.       [simple ageement]
      b.  [The committee] have not yet come to a decision.        [plural override]

Both [2a] and [2b] are acceptable, especially in British English (in American English, the plural override is less likely). According to CGEL (p. 502),

The optionality of the override with collectives reflects the fact that there is potentially a difference of meaning between the versions with singular and plural verbs. From one perspective a committee is a single entity, but since a committee (normally) consists of a plurality of members it can be conceptualised as denoting this plural set. The construction with a plural verb focuses on the members of the committee rather than on the committee as a unit.

Besides committee, staff, and family, there are other clear examples of collective singular nouns that allow optional plural override (with appropriate verbs), such as administration, army, board, class (in one of its meanings), clergy, crew, enemy, etc. A list is given in CGEL, p. 503.

In short: in your examples, both the singular verb is and the plural verb are are acceptable. In the case of staff, if you use the plural, you are thinking of staff as a set consisting of a plurality of members. True, staff is merely the oblique (the complement of of), and is not the head of the NP which is the subject of the sentence; rather, rest is the head. But rest is number-transparent, and so the number of the whole NP is determined by the number of the oblique. And, as we said, the number of the oblique is, through the (optional) plural override, taken to be plural. In contrast, if you use the singular verb, you are thinking of staff as a single entity, and now it is this singular number that 'percolates up' to determine the number of the whole NP. The case of family works just the same.